Degrees of freedom in the energy density formula for black body radiation

I'm looking for a conceptual explanation of a formula in Quantum Mechanics Demystified introduction. They introduce you to the theoretical black body radiation experiment, where demonstrated how a classical approach leads to the ultraviolet catastrophe.

In the explanation they have the following formula for energy density u([itex]\nu[/itex],t):

My understanding is that that degrees of freedom is the number of variables that you can vary in the formula.

Since the degrees of freedom vary, I'm assuming that under certain temperatures and frequencies, the formula can be simplified and reduced? This would in turn change the degrees of freedom? Please let me know if I'm understanding it correctly.